Pairing-based cryptography

Chapter 23 showed us how to build DL systems on the Jacobian of curves. In Chapter 1 we introduced DL systems with bilinear structure. In this chapter we first give more applications of this construction, namely the extension of the tripartite protocol given before to multiparty key exchange, identity-based cryptography, and short signatures. In recent years many systems using this extra structure have been proposed. We include some more references to further work in the respective sections, since giving a complete survey of all these schemes is completely out of the scope of this book. For a collection of results on pairings we refer to the "Pairing-Based Crypto Lounge" [BAR]. The second section is devoted to realizations of such systems. In Chapter 6 we gave the mathematical theory for the Tate–Lichtenbaum pairing and Chapter 16 provided algorithms for efficient evaluation of this pairing on elliptic curves and the Jacobian of hyperelliptic curves. There we assumed that the embedding degree (i.e., the degree k of the extension field Fqk to which the pairing maps), is small, so as to guarantee an efficiently computable map as required in a DL system with bilinear structure. In Section 24.2 we explain for which curves and fields these requirements can be satisfied and give constructions